stick意思是"伸"，stick one's neck out就是伸出脖子来。这个习惯用语很可能来自宰鸡的时候鸡伸长脖子叫唤，反倒正好给人下刀的机会，引申为“冒险、担风险”。

【例句】
He was only too pleased to let someone else stick their neck out and take responsibility.
他太乐意让别人出头冒风险，担责任。

During my political life I've earned myself a reputation as someone who'll stick his neck out, a bit of a rebel.
我在政治生涯中敢冒风险、有点叛逆是小有名气的。